81 research outputs found

### Commutator estimates on contact manifolds and applications

This article studies sharp norm estimates for the commutator of
pseudo-differential operators with multiplication operators on closed
Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate:
If $D$ is a first-order operator in the Heisenberg calculus and $f$ is
Lipschitz in the Carnot-Caratheodory metric, then $[D,f]$ extends to an
$L^2$-bounded operator. Using interpolation, it implies sharp weak--Schatten
class properties for the commutator between zeroth order operators and H\"older
continuous functions. We present applications to sub-Riemannian spectral
triples on Heisenberg manifolds as well as to the regularization of a
functional studied by Englis-Guo-Zhang.Comment: 31 pages, improved presentation and additional reference

### On a preconditioner for time domain boundary element methods

We propose a time stepping scheme for the space-time systems obtained from
Galerkin time-domain boundary element methods for the wave equation. Based on
extrapolation, the method proves stable, becomes exact for increasing degrees
of freedom and can be used either as a preconditioner, or as an efficient
standalone solver for scattering problems with smooth solutions. It also
significantly reduces the number of GMRES iterations for screen problems, with
less regularity, and we explore its limitations for enriched methods based on
non-polynomial approximation spaces.Comment: 15 pages, 16 figure

### Stability Analysis in Magnetic Resonance Elastography II

We consider the inverse problem of finding unknown elastic parameters from
internal measurements of displacement fields for tissues. In the sequel to
Ammari, Waters, Zhang (2015), we use pseudodifferential methods for the problem
of recovering the shear modulus for Stokes systems from internal data. We prove
stability estimates in $d=2,3$ with reduced regularity on the estimates and
show that the presence of a finite dimensional kernel can be removed. This
implies the convergence of the Landweber numerical iteration scheme. We also
show that these hypotheses are natural for experimental use in constructing
shear modulus distributions.Comment: 14 page

### Space-time adaptive finite elements for nonlocal parabolic variational inequalities

This article considers the error analysis of finite element discretizations
and adaptive mesh refinement procedures for nonlocal dynamic contact and
friction, both in the domain and on the boundary. For a large class of
parabolic variational inequalities associated to the fractional Laplacian we
obtain a priori and a posteriori error estimates and study the resulting
space-time adaptive mesh-refinement procedures. Particular emphasis is placed
on mixed formulations, which include the contact forces as a Lagrange
multiplier. Corresponding results are presented for elliptic problems. Our
numerical experiments for $2$-dimensional model problems confirm the
theoretical results: They indicate the efficiency of the a posteriori error
estimates and illustrate the convergence properties of space-time adaptive, as
well as uniform and graded discretizations.Comment: 47 pages, 20 figure

### On the magnitude function of domains in Euclidean space

We study Leinster's notion of magnitude for a compact metric space. For a
smooth, compact domain $X\subset \mathbb{R}^{2m-1}$, we find geometric
significance in the function $\mathcal{M}_X(R) = \mathrm{mag}(R\cdot X)$. The
function $\mathcal{M}_X$ extends from the positive half-line to a meromorphic
function in the complex plane. Its poles are generalized scattering resonances.
In the semiclassical limit $R \to \infty$, $\mathcal{M}_X$ admits an asymptotic
expansion. The three leading terms of $\mathcal{M}_X$ at $R=+\infty$ are
proportional to the volume, surface area and integral of the mean curvature. In
particular, for convex $X$ the leading terms are proportional to the intrinsic
volumes, and we obtain an asymptotic variant of the convex magnitude conjecture
by Leinster and Willerton, with corrected coefficients.Comment: 20 pages, 3 figures, to appear in American Journal of Mathematic

### A deterministic optimal design problem for the heat equation

For the heat equation on a bounded subdomain $\Omega$ of $\mathbb{R}^d$, we
investigate the optimal shape and location of the observation domain in
observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat
packets allows us to remove the randomisation procedure and assumptions on the
geometry of $\Omega$ in previous works. The explicit nature of the heat packets
gives new information about the observability constant in the inverse problem.Comment: 22 page

### Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

We consider the spectral behavior and noncommutative geometry of commutators
$[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a
multiplication operator by a function. When $f$ is H\"{o}lder continuous, the
spectral asymptotics is governed by singularities. We study precise spectral
asymptotics through the computation of Dixmier traces; such computations have
only been considered in less singular settings. Even though a Weyl law fails
for these operators, and no pseudo-differential calculus is available,
variations of Connes' residue trace theorem and related integral formulas
continue to hold. On the circle, a large class of non-measurable Hankel
operators is obtained from H\"older continuous functions $f$, displaying a wide
range of nonclassical spectral asymptotics beyond the Weyl law. The results
extend from Riemannian manifolds to contact manifolds and noncommutative tori.Comment: 40 page

### Adaptive FE-BE coupling for strongly nonlinear transmission problems with friction II

This article discusses the well-posedness and error analysis of the coupling
of finite and boundary elements for transmission or contact problems in
nonlinear elasticity. It concerns W^{1,p}-monotone Hencky materials with an
unbounded stress-strain relation, as they arise in the modelling of ice sheets,
non-Newtonian fluids or porous media. For 1<p<2 the bilinear form of the
boundary element method fails to be continuous in natural function spaces
associated to the nonlinear operator. We propose a functional analytic
framework for the numerical analysis and obtain a priori and a posteriori error
estimates for Galerkin approximations to the resulting boundary/domain
variational inequality. The a posteriori estimate complements recent estimates
obtained for mixed finite element formulations of friction problems in linear
elasticity.Comment: 20 pages, corrected typos and improved expositio

### A Nash-Hormander iteration and boundary elements for the Molodensky problem

We investigate the numerical approximation of the nonlinear Molodensky
problem, which reconstructs the surface of the earth from the gravitational
potential and the gravity vector. The method, based on a smoothed
Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems
and uses a regularization based on a higher-order heat equation to overcome the
loss of derivatives in the surface update. In particular, we obtain a
quantitative a priori estimate for the error after m steps, justify the use of
smoothing operators based on the heat equation, and comment on the accurate
evaluation of the Hessian of the gravitational potential on the surface, using
a representation in terms of a hypersingular integral. A boundary element
method is used to solve the exterior problem. Numerical results compare the
error between the approximation and the exact solution in a model problem.Comment: 32 pages, 14 figures, to appear in Numerische Mathemati

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